Note that here we have two limits. In the limit \(N_{b}\rightarrow +\infty \) ,
the estimate \(\hat{\lambda }(k,\tau _{b},N_{b})\)  converges to \(\lambda (k,\tau _{b})=\frac{1}{\tau _{b}}\log \mathbb {E}\left[e^{k_{0}^{\tau _{b}}A(X(t))\textrm {d}t}\right],\) 
which, in the limit \(\tau _{b}\rightarrow +\infty \) , converges to the
scaled cumulant generating function \(\lambda (k)\)  (and consequently
the same holds for the convergence of \(\hat{I}(k,\tau _{b},N_{b})\) 
to \(I(a)\) ). The convergence of both limits has to be checked to ensure
the correct computation of the large deviation functions. In a practical
application one is constrained by the fixed length \(T=N_{b}\tau _{b}\) 
of the time-series, so that one faces a trade off in the choice of
\(\tau _{b}\)  and \(N_{b}\) . The appeal of this method over attempting
at a direct estimate of the rate function is that one can check precisely
the convergence, exploiting results on the convergence of estimators
of expectation values of exponentials of sums of random variables
[1]}. The convergence is limited by two problems.
